For the sake of completeness, let us briefly recall the construction of the random walk process on a hypergraph and invite the interested reader to consult [1]} for further details. The agents are located on the nodes and hop between them. In a general setting, the walkers
may weight hyperedges depending on their size, introducing a bias in their moves that we shall encode into a function \(\varphi \)  of the hyperedge size. This yields the weighted adjacency matrix \(\mathbf {D}=\mathbf {e}\,\Phi \,\mathbf {e}^{\top }\) , already defined in Eq. (REF ) and hereby recalled:
\(D_{ij} = \sum _{\alpha } e_{i\alpha }\Phi _{\alpha \alpha }e_{j\alpha }\quad \forall i\ne j\text{ and }D_{ii}=\varphi (1) \, ,\) 
